Definition:Special Linear Group

Definition

Let $R$ be a commutative ring with unity whose zero is $0$ and unity is $1$.

The special linear group of order $n$ on $R$ is the set of square matrices of order $n$ whose determinant is $1$.

It is a group under (conventional) matrix multiplication.

It is denoted $\SL {n, R}$, or $\SL n$ if the ring is implicit.

The ring itself is usually a standard number field, but can be any commutative ring with unity.

Also denoted as

Some authors prefer $\map {\mathrm {SL}_n } R$ and $\operatorname{SL}_n$ for the special linear group over $\SL {n, R}$.

Also see

• Special Linear Group is Normal Subgroup of General Linear Group

• Results about the special linear group can be found here.

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 36$. Subgroups
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): general linear group
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): general linear group